In probability and statistics the negative binomial distribution is a discrete probability distribution. It can be used to describe the distribution arising from an experiment consisting of a sequence of independent trials, subject to several constraints. Firstly each trial results in success or failure, the probability of success for each trial, p, is constant across the experiment and finally the experiment continues until a fixed number of successes have been achieved.

The Pascal distribution and the Polya distribution are special cases of the negative binomial. There is a convention among engineers, climatologists, and others to reserve "negative binomial" in a strict sense or "Pascal" (after Blaise Pascal) for the case of an integer-valued parameter r, to the right, and use "Polya" (for George Pólya) for the real-valued case. The Polya distribution more accurately models occurrences of "contagious" discrete events, like tornado outbreaks, than does the Poisson distribution.

Specification of the negative binomial distribution

Probability mass function

The family of negative binomial distributions is a two-parameter family; several parametrizations are in common use. One very common parameterization employs two real-valued parameters p and r with 0 < p < 1 and r > 0. Under this parameterization, the probability mass function of a random variable with a NegBin(r, p) distribution takes the following form:

f(k;r,p) = {k+r-1 \choose k}\cdot p^r \cdot (1-p)^k \!

for k = 0,1,2,...

where

{k+r-1 \choose k} = \frac{\Gamma(k+r)}{k!\cdot\Gamma(r)} = (-1)^k\cdot{-r \choose k}\!

and

\Gamma(r) = (r-1)!

Limiting case

Under an alternative parameterization

\lambda = r\cdot(p^{-1}-1) \!

p = \frac{r}{r+\lambda} \!

the mass function becomes

g(k) = \frac{\lambda^k}{k!} \cdot \frac{\Gamma(r+k)}{\Gamma(r)\;(r+\lambda)^k} \cdot \frac{1}{\left(1+\frac{\lambda}{r}\right)^{r}} \!

where ? and r are nonnegative real parameters. Under this parameterization, we have

\lim_{r\to\infty} g(k) = \frac{\lambda^k}{k!} \cdot 1 \cdot \frac{1}{\exp(\lambda)} \!

which is the mass function of a Poisson-distributed random variable with Poisson rate ?. In other words, the alternatively parameterized negative binomial distribution converges to the Poisson distribution and r controls the deviation from the Poisson. This makes the negative binomial distribution suitable as a robust alternative to the Poisson, which approaches the Poisson for large r, but which has larger variance than the Poisson for small r.

Gamma-Poisson mixture

Third, the negative binomial distribution arises as a continuous mixture of Poisson distributions where the mixing distribution of the Poisson rate is a gamma distribution. Formally, this means that the mass function of the negative binomial distribution can also be written as